Namely, do any of them overlap with one another? To give an example, does existential quantification apply to counterfactual statements?

A counterfactual statement can be something like "A possible world where I won the lottery." I wouldn't be able to assign an existential quantifier in the same manner that I could to a statement in this world, because that statement simply does not obtain or isn't instantiated in this world contrary to a possible world where I did indeed win the lottery.

Im in the middle of writing an answer to your question but I want to make sure before I finish it, the question that you are asking is whether or not first order modal logic exists, right? As in you are asking whether or not there is an established literature detailing the development and use of these types of systems? And then maybe also, accepting that they exist, are you asking how the existential quantifier is interpreted in those systems?
– Not_HereDec 16 '18 at 15:08

Yes, I'm mainly interested in how is the existential quantifier interpreted in those systems.
– WallowsDec 16 '18 at 23:03

1 Answer
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Such systems are called first-order modal logic. There is no single accepted method how to handle quantifying into modal contexts. In fact, it will depend on your metaphysical standpoint; for more details I refer you to the SEP article intensional and extensional contexts.